All Questions
Tagged with galois-representations reference-request
57 questions
3
votes
0
answers
90
views
Reference request: étale local system on $\Gamma\backslash\mathcal H$ for $\Gamma$ non-congruence
Suppose $\mathcal H$ is the upper half plane, and $\Gamma\subset \operatorname{PSL}_2(\mathbb Z)$ is a finite index subgroup. Then by Belyi's theorem we know that $\Gamma\backslash\mathcal H$ is an ...
- 775
4
votes
0
answers
127
views
mod $p$ local Galois representation attached to elliptic curves
In the paper, lemma 4.4. The author gives the form of the representation of $G_p$ on $E[p]$ of the form
$$\begin{pmatrix} \varepsilon\chi & *\\0 & \chi^{-1} \end{pmatrix}.$$
Do they assumed ...
- 275
2
votes
1
answer
186
views
On local Galois deformation rings
Let $p,\ell$ be two different primes. Let $K$ be a finite field extension of $\mathbb{Q}_{\ell}$ and $ \bar{\rho}:G_{K}\to {\rm GL}_{n}(\mathbb{F}_p) $ be a continuous mod $p$ representation of the ...
- 667
1
vote
0
answers
98
views
Are there known effective bounds on the number of semisimple Galois representations?
In continuation to my question here, are there known effective bounds on the total number of semisimple $p$-adic Galois representations unramified outside a finite set of primes $S$, of dimension $d$, ...
- 2,907
0
votes
1
answer
223
views
Reference for Faltings' proof on finiteness of semisimple $d$-dimensional $p$-adic Galois representations
I'm looking for a reference to Faltings' proof concerning the finiteness of $d$-dimensional semisimple $p$-adic Galois representations. Specifically, the result states that there are only finitely ...
- 2,907
3
votes
1
answer
249
views
Action of complex conjugation on etale cohomology
Let $X$ be a genus $g$ smooth projective curve, defined over $\mathbb{Q}$, and let $\overline{X}$ denote the base change of $X$ to $\overline{\mathbb{Q}}$.
It is well known that $H^1_{\text{ét}}(\...
- 2,907
5
votes
1
answer
228
views
Lifting mod $p$ representations of arithmetic fundamental groups of a non-affine scheme over a finite field of characteristic $p$
Let $X$ be a geometrically irreducible scheme (not necessarily affine) over $\mathbb{F}_{p}$ and let $ \pi_{1}(X) $ be the arithmetic etale fundamental group of $ X $. Let $ \overline{\mathbb{F}}_{p} $...
- 863
1
vote
0
answers
47
views
Absolute irreducibility implies free action on framed universal deformation ring
Let $\overline{\rho}: G\longrightarrow \text{GL}_n(\mathbb{F}_p)$ be a residual representation of the Galois group of a number field. Let $R_{\overline{\rho}}^{\square,\text{univ}}$ be the universal ...
- 2,907
1
vote
0
answers
127
views
Multiplicities of Galois representations in the semisimplification of the reduction of a Tate module
Let $C$ be a smooth proper curve, of genus $g$ over a number field $K$. Let $v$ be a prime of good reduction for $C$ above $p>2$, and let $T_pJ$ denote the $p$-adic Tate module of $J$, the Jacobian ...
- 2,907
1
vote
0
answers
131
views
Analytic properties of $L$-functions attached to a compatible system of $\ell$-adic Galois representations
Let $F$ and $E$ be number fields, $G_F$ be the absolute Galois group of $F$, and $S$ be a finite set of primes of $F$. For $\lambda$ a prime of $E$ we denote by $\ell$ its residual characteristic. We ...
- 663
3
votes
0
answers
122
views
Description of $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for a CM elliptic curve $E$
I am looking for a specific description of the Galois groups $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for an elliptic curve $E$ with complex ...
- 309
3
votes
0
answers
145
views
Reference showing no proper subgroups of p-adic orthogonal groups surject onto mod p orthogonal groups
I am looking for a reference for the following statement:
Let $O$ be an orthogonal group associated to a nondegenerate quadratic form of rank $r$ over the p-adic integers $\mathbb Z_p$. Suppose $r$ is ...
- 1,308
2
votes
1
answer
243
views
$\pi$-adic Galois representations of attached to newforms at $p \nmid N$ are crystalline
Is [Scholl, Motives for modular forms, Theorem 1.2.4 (ii)] proven for any $p$ independent of the weight?
Concretely, let $f$ be a normalized eigenform of weight $w$. Let $p$ be a prime not dividing ...
user471019
37
votes
1
answer
1k
views
What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
Variants have been asked here before (e.g. Which small finite ...
- 2,959
2
votes
0
answers
229
views
Reference Request - New proof of Ribet's level lowering by Khare and Wintenberger
I'm currently following the note of Sug Woo Shin's course at Berkeley with notes taken by Rong Zhou. In Section 24.3 (Page 86), Ribet's level lowering theorem is stated:
[Theorem 24.7] $E = E_{a^{\...
- 639
3
votes
0
answers
152
views
Finiteness of points over the cyclotomic extension for modular forms
Let $\rho(f):G_\mathbb{Q} \rightarrow GL_2(K_f)$ be the Galois representation attached to some cuspidal modular form $f$ where $K_f$ is a finite extension of $\mathbb{Q}_p$.
Let $V_f$ be the vector ...
- 313
3
votes
0
answers
131
views
Where general mixed Galois representations are defined?
I am interested in etale cohomology of varieties, and respectively, in mixed $\mathbb Q_{\ell}$-adic Galois representations over finitely generated fields. What is the canonical reference for this ...
- 16.9k
4
votes
1
answer
412
views
Motivations of families of modular forms, elliptic curves and Galois representations?
I want to know some reference, why do some number theorists study the families of the elliptic curves, modular forms or Galois representations? As far as I know, I always consider the Galois ...
user141691
6
votes
0
answers
337
views
Stacks project for Galois representations and automorphic forms
Is there anything like Stacks project for Galois representations and automorphic forms? I am not asking people to start something like Stacks project, just asking if something like Stacks project ...
3
votes
1
answer
171
views
Congruence of normalized eigenforms at two primes
Let $f_i\in S_{k_i}(\Gamma_0(N_i))$ be normalized cuspidal eigenforms for $i=1,2$ and let $K$ be the composite of the fields of Fourier coefficients generated by $f_1$ and $f_2$ and let $\mathfrak{p}...
user130124
7
votes
0
answers
379
views
Local properties of Galois representations attached to torsion classes
$\DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\GL}{GL} \newcommand{\F}{\mathbb{F}} \newcommand{\p}{\mathfrak{p}} \DeclareMathOperator{\Sym}{Sym}$
Let $F$ be a number field, and let $\Gamma$ be ...
- 5,382
2
votes
0
answers
154
views
Categorical representations of absolute Galois groups
I am looking for interesting examples of categorical representations of absolute Galois groups of arithmetic fields. Pointers to the literature would be appreciated.
user127738
11
votes
1
answer
1k
views
Reference request: Newton above Hodge
Let $K$ be a p-adic field, and let $\mathcal{O}$ be the ring of integers inside $K$ with residue field $k$. Let $\mathcal{X}$ be a smooth proper formal scheme over $\mathcal{O}$ (with topology given ...
- 619
9
votes
3
answers
2k
views
Finiteness of Galois cohomology
Let $k$ be a field, $X$ a smooth projective variety over $k$, $\overline{X} := X\times_k {k}^{\rm sep}$ for a separable closure ${k}^{\rm sep}$ of $k$, $\ell$ a prime with $\ell\in k^{\times}$.
Are ...
user92332
5
votes
0
answers
169
views
Where does the notation $\operatorname{Tr}(\cdot)\bmod \ell^\alpha$ implies isomorphism come from?
In J-P Serre's article on Faltings-Serre (Resume du Course 1984-1985) he states (without proof) that for two finite-dimensional $\ell$-adic Galois representations of $\operatorname{Gal}(\mathbb{Q})$, ...
- 2,429
10
votes
2
answers
4k
views
Reference book for Galois Representations
I am an undergrad. I have taken courses in algebraic number theory and have a basic idea about $p$-adic numbers. I have also read a little bit of infinite Galois theory. But I have no idea about ...
- 239
6
votes
1
answer
380
views
Applications of Level Lowering
What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other ...
- 1,629
5
votes
2
answers
402
views
Is Howe's construction of tame supercuspidal representations independent of additive character?
Let $F$ be a $p$-adic field.
In "Tamely ramified supercuspidal representations of $Gl_n$" (Am. J. Math 73 (1977)), Howe constructs a supercuspidal representation $\pi_{\psi}$ of $GL_n(F)$ from the ...
- 1,453
5
votes
1
answer
441
views
Mazur's Galois Deformations paper for non-residually irreducible case
In Barry Mazur's paper introducing Galois deformations, he hints at having a general theory for representations which are not residually Schur, but with more complicated statements. Does anyone know ...
- 2,429
1
vote
1
answer
881
views
Why is every l-adic Galois representation conjugate to one over the l-adic integers? [closed]
Why is every l-adic Galois representation
$$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Q}_{l})$$
conjugate to one over the l-adic integers?
$$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Z}_{l})$$
- 1,805
7
votes
1
answer
646
views
Serre's surjective theorem importance
I'm studying Serre's paper in wich he shows the following theorem:
Let K be a number field, $E$ an elliptic curve over K without CM. Then the representation $$\rho_{\ell}:\mathrm{Gal}(\bar K/K)\...
- 205
2
votes
1
answer
223
views
Level-Lowering in Weight 2
Let $N$ and $p$ be relatively prime integers with $p$ a prime. Suppose $f$ is a weight $k=2$ (normalized, cuspidal, etc) newform of level $\Gamma_1(N) \cap \Gamma_0(p)$. I seem to recall the existence ...
- 1,422
1
vote
1
answer
291
views
Explicit deformations of pseudo representations
Let $G$ be a group (which I will be glad to consider to be the absolute Galois group of a $p$-adic field, and so satisfies Mazur's finiteness condition which appears in his paper Deforming Galois ...
- 113
7
votes
0
answers
313
views
Any counterexamples known for the Generalized Tate conjecture?
One can state the generalized Tate conjecture over arbitrary finitely generated fields; to this end one should just define Galois representation to be effective if the eignevalues of the actions of ...
- 16.9k
5
votes
0
answers
585
views
Bloch Kato Exponential as formal lie group exponential
Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism
$$DR(V) \...
- 3,555
13
votes
0
answers
282
views
Is there literature on approaching semisimplicity of l-adic cohomology using vanishing cycles
Let $K$ be a finitely generated field, and $X/K$ a smooth proper variety. Let $G$ denote the absolute Galois group $\mathrm{Gal}(\bar{K}/K)$. Let $\ell$ be a prime different from $\mathrm{char}(K)$. A ...
- 5,494
8
votes
0
answers
335
views
Irreducibility of Galois representations attached to unitary groups
If $G$ is a unitary group in $n$ variables over $\mathbb Q$, attached to an hermitian form for an imaginary quadratic extension $E/\mathbb Q$ and if we suppose that the hermitian form is definite over ...
- 26k
5
votes
1
answer
842
views
Reference for $p$-adic Hodge theory with coefficients
Let $K$ be a $p$-adic field and $L$ be a finite or infinite extension (maybe algebraic ?) of $\mathbb{Q}_p$.
Is there a reference for $p$-Hodge theory for representations $\rho : Gal_K \rightarrow ...
- 527
4
votes
2
answers
450
views
Is there a version of Serre's modularity conjecture for projective representations?
Serre's modularity conjecture asserts that a continuous odd irreducible representation $$\overline{\rho} : G_\mathbb{Q} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$$ must be modular, in the sense that ...
- 3,142
3
votes
1
answer
389
views
Galois deformations with Panchiskin condition
Let $L/\mathbf{Q}_p$ be a finite extension and we consider a fixed $L$-linear representation $V$ of the absolute Galois group $G:=\operatorname{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$. Assume that ...
- 63
2
votes
1
answer
338
views
Finite Flat Group Schemes for Modular Forms of Higher Weight
Let $f$ be a newform (normalized, cuspidal) of weight $k \ge 2$. Then for a prime $\ell$ there is an $\ell$-adic Galois representation associated to $f$. If $k=2$, this comes from an abelian variety, ...
- 15.4k
5
votes
0
answers
288
views
Do infinite and ramified local factors of the Dedekind zeta function of a tame number field characterize its local root numbers?
Let say you have two number fields, that are tamely ramified, and suppose that the $p$-part of their Dedekind zeta functions coincide for all prime $p$ which is ramified in either field. Suppose ...
- 2,396
31
votes
1
answer
5k
views
Modern proof of Serre's open image theorem?
Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic Curves'...
- 1,905
1
vote
0
answers
162
views
Construction of RM abelian variety from eigenform
Let $f$ be a normalized eigenform of weight $2$ level $N$. If the Fourier coefficients of $f$ generate a totally real field $F$, then we associate to $f$ a system of $\ell$-adic Galois representations ...
- 15.4k
1
vote
0
answers
231
views
Where can I find a copy of Serre's Cours au college de France 1985-1986?
Hi,
I was wondering: where might I be able to find a copy of this work online?
And are there any other resources for the proof of the open image theorem for abelian varieties with endomorphism ring ...
- 15.4k
13
votes
1
answer
1k
views
Reference for: CM Hilbert Modular forms arise from Hecke characters
For classical modular forms, the correspondence between the form having CM by an imaginary quadratic field $K$ and it being induced from a Hecke character on $K$ is well-known. (Ribet's paper is a ...
- 659
6
votes
1
answer
1k
views
Mumford-Tate group and Galois representations
Could someone please point me towards a proof of why the image of a Galois representation on the Tate-module of an abelian variety is limited by its Mumford-Tate group?
- 1,905
16
votes
0
answers
11k
views
Deligne's letter to Jean-Pierre Serre
I'm looking for another letter of Pierre Deligne, this time to Jean-Pierre Serre (from around 1974 I think), in which he proves that the Galois representation associated to a certain Hecke eigenform ...
8
votes
1
answer
464
views
Image of complex conjugation by modular representations in characteristic 2
The question I am going to ask looks well-known, and I even may have heard things about it (but since I used to be deaf to anything in characteristic 2, whatever I heard has never been recorded in my ...
- 26k
41
votes
2
answers
17k
views
Introductory text on Galois representations
Could someone please recommend a good introductory text on Galois representations? In particular, something that might help with reading Serre's "Abelian l-Adic Representations and Elliptic Curves" ...